Step 1 :
Divide $ 17690 $ by $ 12690 $ and get the remainder

$$ 17690 : 12690 = 1 \text{ ( remainder is } \color{blue}{ 5000 } \text{ ) } $$

The remainder is positive ($ 5000 > 0 $), so we will continue with division.

Step 2 :
Divide $ 12690 $ by $ \color{blue}{ 5000 } $ and get the remainder

$$ 12690 : 5000 = 2 \text{ ( remainder is } \color{blue}{ 2690 } \text{ ) } $$

The remainder is still positive ($ 2690 > 0 $), so we will continue with division.

Step 3 :
Divide $ 5000 $ by $ \color{blue}{ 2690 } $ and get the remainder

$$ 5000 : 2690 = 1 \text{ ( remainder is } \color{blue}{ 2310 } \text{ ) } $$

The remainder is still positive ($ 2310 > 0 $), so we will continue with division.

Step 4 :
Divide $ 2690 $ by $ \color{blue}{ 2310 } $ and get the remainder

$$ 2690 : 2310 = 1 \text{ ( remainder is } \color{blue}{ 380 } \text{ ) } $$

The remainder is still positive ($ 380 > 0 $), so we will continue with division.

Step 5 :
Divide $ 2310 $ by $ \color{blue}{ 380 } $ and get the remainder

$$ 2310 : 380 = 6 \text{ ( remainder is } \color{blue}{ 30 } \text{ ) } $$

The remainder is still positive ($ 30 > 0 $), so we will continue with division.

Step 6 :
Divide $ 380 $ by $ \color{blue}{ 30 } $ and get the remainder

$$ 380 : 30 = 12 \text{ ( remainder is } \color{blue}{ 20 } \text{ ) } $$

The remainder is still positive ($ 20 > 0 $), so we will continue with division.

Step 7 :
Divide $ 30 $ by $ \color{blue}{ 20 } $ and get the remainder

$$ 30 : 20 = 1 \text{ ( remainder is } \color{blue}{ 10 } \text{ ) } $$

The remainder is still positive ($ 10 > 0 $), so we will continue with division.

Step 8 :
Divide $ 20 $ by $ \color{blue}{ 10 } $ and get the remainder

$$ 20 : \color{blue}{ \boxed { 10 }} = 2 \text{ ( remainder is } \color{blue}{ 0 } \text{ ) } $$

The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 10 }} $.

This solution can be visualized using a Venn diagram.

The GCD equals the product of the numbers at the intersection.

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